Calculating the probability of a coin falling on its side

Question

A classical example that's given for probability exercises is coin flipping. Generally it is accepted that there are two possible outcomes which are heads or tails. However, it is possible in the real world for a coin to also fall on its side which makes a third event ( $P(\text{side}) = 1 - P(\text{heads}) - P(\text{tails})$ ?). How would a mathematician go about calculating the probability of that? Could it be done by simply using respective surface areas or would a proper model be more complex?

Answer 1

I covered this question at my Fair Dice column. If you flip your coin onto sticky pitch, it has a much better chance of landing on edge than if you flip it onto a smooth granite surface. The potential to bounce makes a huge difference.

Roughly, you're looking at a 3-state object. Each time it lands, it will lose some energy, and the chance that it will change to a different state varies with the energy. The amount of energy necessary to topple from one state to another will let you build a transition matrix, and then multiply a series of these together to get expected behavior on a given surface.

More is at the site dicephysics. Cylinders of various size were mechanically rolled thousands of times on a variety of surfaces. Results closely followed the Energy State model of my thesis on dice.

Toppling must be considered. Many objects have unstable sides, and will topple away from them. There is a Unistable polyhedron which will always eventually roll to the same face each time it it tossed.

Answer 2

Experiments and models of physical coin flipping are given in:

Diaconis, Holmes, and Montgomery, Dynamical Bias in the Coin Toss comptop.stanford.edu/preprints/heads.pdf

and

Murray and Teare, The probability of a tossed coin falling on its edge, Phys. Rev. E. 2547-2552 (1993)

abstract of 1993 paper: "An experiment is reported in which an object which can rest in multiple stable configurations is dropped with randomized initial conditions from a height onto a flat surface. The effect of varying the object’s shape on the probability of landing in the less stable configuration is measured. A dynamical model of the experiment is introduced and solved by numerical simulations. Results of the experiments and simulations are in good agreement, confirming that the model incorporates the essential features of the dynamics of the tossing experiment. Extrapolations based on the model suggest that the probability of an American nickel landing on edge is approximately 1 in 6000 tosses."

Answer 3

While I am and always have been a big proponent of the edge factoring into probability in just a quirky enough way as to disrupt an absolute $50-50$ "fair" game and create the equivalent to the least profitable house advantage in existence kind of like $0$ and $00$ on a roulette wheel, the idea that the probability is as good as $1$ in $6000$ or $.0166%$ seems kind of high even for a nickel.

I guess I think about it in terms of if possibility exists in any way, regardless of probability, in the moment of the flip is the coin in a state of fluctuation that is suggestive of one of $2$ outcomes but indeed it is not a definite zero-sum game in that a $3$rd party, as moronic as his speculation may be, could be the benefactor of the outcome should he place his position on a "house" win... I dunno, it's that kind of stuff that distracts me from learning anything productive! :)